Algebraic Reduction and Periodic Solvability in a Coupled Ternary Rational System

In this paper, we investigate a new class of three-component nonlinear rational difference equations of the second order characterized by structured periodic interactions. Through a carefully designed algebraic transformation and the introduction of suitable auxiliary sequences, the original nonlinear model is converted into an equivalent periodic scheme of order six. This reformulation enables the complete determination of explicit solution formulas in closed form. We establish precise conditions under which the solutions remain well defined and analytically tractable. A series of illustrative numerical experiments reveals a wide spectrum of dynamical behaviors, ranging from oscillatory patterns to various modes of convergence.